The ring ${\mathrm{M}}_{8k+4}({\mathbb{Z}}_2)$ is nil-clean of index four

Abstract

We show that the direct sum of an odd number of matrices $$C=\left(\begin{array}{cccc}0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 1\end{array}\right)$$ cannot be a sum $P+Q$ of matrices over ${\mathbb{F}}_2$ satisfying $P^2=P$ and $Q^3=O$.

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